Reach the Hargreave-Rasch building
Story related. When you exit the water and enter the Hargreaves building in chapter 11, Corporate Collapse, you will get this.
Create your first custom class
As easy as they get. In the multiplayer, go into your custom classes and customize your weapon with an attachment or camo and then exit out. Once you are back at the main screen, you will receive the achievement/trophy.
Infiltrate the CELL facility at Wall Street
Story related. Received during chapter 7, Dead Man Walking.
Infiltrate the alien hive
Story related. Received during chapter 9, Dark Heart.
Multiplayer: Win a match of every game mode
You will need to play and win one game in each of the 6 game modes found in the game. To unlock all of them, you will need to reach level 18 first. The modes are:
Capture the Relay
Team Instant Action
Multiplayer: Finish top of the Scoreboard
Come in first place in a Multiplayer match.
Scan all of Richard Morgan's books in the NY public library
After you enter the library, you will need to use the Maximum armor to pass through a fire filled doorway. Head to the right and there will be book shelves there that need to be scanned. Equip the visor and then press X/Square to scan them and get this easy reward.
Multiplayer: Fully level a Suit Module
Instead of having to fully upgrade the Nanosuit through XP as in the “Dressed to Kill” achievement/trophy, you need to fully level one of the modules within the three categories to level 3 that can be found in the Operative Status/Skill Assessments screen. There are numerous modules for each category, but only one needs to be satisfied to earn this.
Complete 12 levels on Supersoldier difficulty
The easiest way is to play through on either Recruit or Soldier first so that you can upgrade your suit to its max. The upgrades carry over through playthroughs so the Supersoldier playthrough will allow you to use those upgrades.